Jabari is power washing houses for a summer job. For every job, he charges an initial fee plus $\$30$ for each hour of work. His total fee for a $4$ -hour job, for instance, is $\$170$. Jabari's total fee, $f$, for a single job is a function of the number, $t$, of hours it takes him to complete the job. Write the function's formula. $f=$
Solution: Jabari's hourly fee is constant, so we are dealing with a linear relationship. We could write the desired formula in slope-intercept form: $f= mt+ b$. In this form, $ m$ gives us the slope of the graph of the function and $ b$ gives us the $y$ -intercept. Our goal is to find the values of $ m$ and $ b$ and substitute them into this formula. We know that Jabari's fee increases at a rate of $\$30$ per hour, so the slope $ m$ is ${30}$, and our function looks like $f={30}t+ b$. We also know that Jabari's fee for a $4$ -hour job is $\$170$, which means that when $t=4$, $f=170$. We can substitute this into the formula of the function to find $ b$ : $\begin{aligned}{30}\cdot4+ b&=170\\\\ 120+ b&=170\\\\ b&={50}\end{aligned}$ This means that Jabari's initial fee is $\$50$ for each job. Since $ m = {30}$ and $ b = {50}$, the desired formula is: $f={30}t+{50}$